Generalization of hypergeometric type differential equation

[cg78ithaca]

I am aware that hypergeometric type differential equations of the type:

can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant, but a 3rd-degree polynomial, and I'm not sure how to solve Mellin-transformed version of the equation above if λ is a polynomial and not a constant, if indeed it is solvable analytically.

Does the equation above have analytical solutions if λ(s) is a 3rd-degree polynomial, and if so, how do I arrive at them?

 

[Paul Colby]

I played around with equations "similar" to these in graduate school. I would just point out (forgive if this is obvious) if $\sigma(s)$ and $\tau(s)$ are restricted to polynomials of second and first order then the equation viewed as an operator on $y(s)$ sends polynomials of order $n$ into polynomials of order $n$ which I believe is what makes them "special". Letting $\lambda$ be a polynomial in $s$ will break this property.